Article submitted to IET Microwaves, Antennas and Propagation: "A necessary dispersion requirement is given for the effective relative permeability, μr (ω) and permittivity ǫr (ω) as a function of frequency for lossy homogeneous dispersive materials and negative refractive index materials in particular, if they exist at some dimensional scale D. The requirements are valid both for isotropic and uniaxial materials, the latter defined by material properties that are invariant under a rotation about a principal axis. Using asymptotic requirements at zero frequency and Foster’s reactance theorem, the general band-gap structure is given for low-loss materials. It is shown that such materials can only be achieved if the material exhibits finite band-gaps separating regions of positive and/or negative refractive index regions. In degenerate cases the band gaps shrink to zero but low-loss propagation remains impossible in the neighbourhood of the critical frequencies. The effect of small losses are considered, and compared with Stockman’s criterion. Losses may be arbitrarily small within a band provided there are no bounds to the size of the real parts of the relative permittivity and permeability at the band edges. A numerical example is given showing the effect of small losses.